Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Nonlinear dimensionality reduction models have made tremendous impacts on image and signal representation learning. However, almost all extant methods assume that real-valued Euclidean representations are rich enough faithfully preserve the data. However, flat geometry is too simplistic to meaningfully reflect the topological structure of many classes of data with cyclic structures. In the first half of this talk, we develop several methods for accurately representing data sampled from several non-euclidean domains: graphs, manifolds, varifolds, and simplicial complexes. A common theme in this section is to look for approaches that sidestep the curse of dimensionality by defining models that rely on the problem's intrinsic dimension rather than the dimension in which the observation is made. In the second part of this talk, we develop strategies for analyzing signals in these domains, intrinsically solving approximation, regression, and classification problems. This analysis is based on developing generalizations of classical techniques such as multiscale bases and overcomplete transforms, as well as neural network-based machine learning techniques.

This is a part of the first Joint CeDAR/UCD4IDS Conference.